Initial curvature of beams and shells

This problem contains basic test cases for one or more Abaqus elements and features.

This page discusses:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/Explicit

Elements tested

B21
B21H
B22
B22H
B23
B23H
B31
B31H
B31OS
B31OSH
B32
B32H
B32OS
B32OSH
B33
B33H

PIPE21
PIPE21H
PIPE22
PIPE22H
PIPE31
PIPE31H
PIPE32
PIPE32H

S4
S4R
S4R5
S8R
S8R5
S9R5
STRI3
STRI65
SC6R
SC8R
CSS8

Problem description

Material:

Linear elastic, Young's modulus = 30 × 10⁶, Poisson's ratio = 0.3.

Boundary conditions:

End A is clamped.

Loading:

$F_{x} = $ 25.0 at end B.

General:

Initial curvature is defined by specifying the direction cosines of the normals at the two ends.

Gauss integration is used for the shell cross-section for the S4R elements.

Reference solution

Reference results are generated from models consisting of 20 B33 cubic beam elements. (Since only one element is used for modeling, if the direction cosines of the normals are not used, the solution will correspond to straight beam theory.) The reference tests use SECTION=RECT, SECTION=I, or SECTION=PIPE. These sections correspond to regular beams and shells, open section beams, and pipes, respectively. Only pipe elements are verified in Abaqus/Explicit.

Regular beams and shells (see erefrrs7.inp):

Displacements, curved beam solution: $u_{x}^{B} = $ 2.1735 × 10⁻⁵, $u_{y}^{B} = $ 1.4570 × 10⁻⁴.
Displacements, straight beam solution: $u_{x}^{B} = $ 1.6667 × 10⁻⁵, $u_{y}^{B} = $ 0.0.

Open section beam elements (see erefois7.inp):

Displacements, curved beam solution: $u_{x}^{B} = $ 3.1946 × 10⁻⁴, $u_{y}^{B} = $ 1.0962 × 10⁻³.
Displacements, straight beam solution: $u_{x}^{B} = $ 2.8153 × 10⁻⁴, $u_{y}^{B} = $ 0.0.

Pipe elements (see erefpps7.inp):

Displacements, curved beam solution: $u_{x}^{B} = $ 2.9461 × 10⁻⁵, $u_{y}^{B} = $ 4.5078 × 10⁻⁵.
Displacements, straight beam solution: $u_{x}^{B} = $ 2.7922 × 10⁻⁵, $u_{y}^{B} = $ 0.0.

Results and discussion

Table 1. Regular beams and shells.
Element Type	$u_{x}^{B}$	$u_{y}^{B}$	Remarks
B21 (1-element mesh)	1.6667 × 10⁻⁵	0.0	Straight*
B21 (Refined mesh)	2.1715 × 10⁻⁵	1.4343 × 10⁻⁴	Curved
B21H (1-element mesh)	1.6667 × 10⁻⁵	0.0	Straight*
B21H (Refined mesh)	2.1665 × 10⁻⁵	1.4343 × 10⁻⁴	Curved
B22	2.1085 × 10⁻⁵	1.4686 × 10⁻⁴	Curved
B22H	2.1085 × 10⁻⁵	1.4686 × 10⁻⁴	Curved
B23	2.0873 × 10⁻⁵	1.4549 × 10⁻⁴	Curved
B23H	2.0873 × 10⁻⁵	1.4549 × 10⁻⁴	Curved
B31 (1-element mesh)	1.6667 × 10⁻⁵	0.0	Straight*
B31 (Refined mesh)	2.1715 × 10⁻⁵	1.4343 × 10⁻⁴	Curved
B31H (1-element mesh)	1.6667 × 10⁻⁵	0.0	Straight
B31H (Refined mesh)	2.1715 × 10⁻⁵	1.4343 × 10⁻⁴	Curved
B32	2.1084 × 10⁻⁵	1.4686 × 10⁻⁴	Curved
B32H	2.1084 × 10⁻⁵	1.4686 × 10⁻⁴	Curved
B33	2.0873 × 10⁻⁵	1.4548 × 10⁻⁴	Curved
B33H	2.0873 × 10⁻⁵	1.4548 × 10⁻⁴	Curved
S4 (1-element mesh)	1.6292 × 10⁻⁵	0.0	Straight*
S4 (Refined mesh)	2.1607 × 10⁻⁵	1.4314 × 10⁻⁴	Curved
S4R (1-element mesh)	1.6666 × 10⁻⁵	0.0	Straight*
S4R (Refined mesh)	2.1661 × 10⁻⁵	1.4340 × 10⁻⁴	Curved
S4R5 (1-element mesh)	1.6666 × 10⁻⁵	0.0	Straight*
S4R5 (Refined mesh)	2.1667 × 10⁻⁵	1.4344 × 10⁻⁴	Curved
S8R	2.1036 × 10⁻⁵	1.4508 × 10⁻⁴	Curved
S8R5	2.1001 × 10⁻⁵	1.4638 × 10⁻⁴	Curved
S9R5	2.1001 × 10⁻⁵	1.4638 × 10⁻⁴	Curved
STRI3	1.6667 × 10⁻⁵	0.0	Straight
STRI65 $†$	2.0750 × 10⁻⁵	1.4331 × 10⁻⁴	Curved
SC6R	2.1673 × 10⁻⁵	1.425 × 10⁻⁴	Curved
SC8R	2.156 × 10⁻⁵	1.2402 × 10⁻⁴	Curved
SC8R	1.6608 × 10⁻⁵	2.5028 × 10⁻⁴	Straight
SC8R**	2.1491 × 10⁻⁵	1.4175 × 10⁻⁴	Curved
SC8R**	1.6276 × 10⁻⁵	2.4271 × 10⁻⁴	Straight
CSS8	2.1369 × 10⁻⁵	1.4093 × 10⁻⁴	Curved
CSS8	1.4511 × 10⁻⁵	1.1925 × 10⁻⁴	Straight

Table 2. Open section beam elements.
Element Type	$u_{x}^{B}$	$u_{y}^{B}$	Remarks
B31OS (1-element mesh)	2.8153 × 10⁻⁴	0.0	Straight*
B31OS (Refined mesh)	3.2287 × 10⁻⁴	1.0790 × 10⁻³	Curved
B31OSH (1-element mesh)	2.8153 × 10⁻⁴	0.0	Straight*
B31OSH (Refined mesh)	3.2287 × 10⁻⁴	1.0790 × 10⁻³	Curved
B32OS (1-element mesh)	3.1787 × 10⁻⁴	1.1048 × 10⁻³	Curved
B32OSH (Refined mesh)	3.1787 × 10⁻⁴	1.1048 × 10⁻³	Curved

Table 3. Pipe elements.
Element Type	$u_{x}^{B}$	$u_{y}^{B}$	Remarks
PIPE21 (1-element mesh)	2.7922 × 10⁻⁵	0.0	Straight*
PIPE21 (Refined mesh)	2.9768 × 10⁻⁵	4.4373 × 10⁻⁵	Curved
PIPE21H (1-element mesh)	2.7922 × 10⁻⁵	0.0	Straight*
PIPE21H (Refined mesh)	2.9768 × 10⁻⁵	4.4373 × 10⁻⁵	Curved
PIPE22 (1-element mesh)	2.9572 × 10⁻⁵	4.5435 × 10⁻⁵	Curved
PIPE22H (1-element mesh)	2.9572 × 10⁻⁵	4.5435 × 10⁻⁵	Curved
PIPE31 (1-element mesh)	2.7922 × 10⁻⁵	0.0	Straight*
PIPE31 (Refined mesh)	2.9768 × 10⁻⁵	4.4373 × 10⁻⁵	Curved
PIPE31H (1-element mesh)	2.7922 × 10⁻⁵	0.0	Straight*
PIPE31H (Refined mesh)	2.9768 × 10⁻⁵	4.4373 × 10⁻⁵	Curved
PIPE32 (1-element mesh)	2.9572 × 10⁻⁵	4.5435 × 10⁻⁵	Curved
PIPE32H (1-element mesh)	2.9572 × 10⁻⁵	4.5435 × 10⁻⁵	Curved

* These are first-order elements and are unable to capture initial curvature with a one-element mesh. However, a refined mesh for these elements yields very good results. $†$ Due to the lack of symmetry for triangular meshes, the displacements at the nodes that are at point B may differ slightly. The maximum values are documented here.** These results are obtained using enhanced hourglass control.

Input files

Coarse mesh tests:

eb22rms7.inp: B21 elements.
eb2hrms7.inp: B21H elements.
eb23rms7.inp: B22 elements.
eb2irms7.inp: B22H elements.
eb2arms7.inp: B23 elements.
eb2jrms7.inp: B23H elements.
eb32rms7.inp: B31 elements.
eb3hrms7.inp: B31H elements.
ebo2ims7.inp: B31OS elements.
ebohims7.inp: B31OSH elements.
eb33rms7.inp: B32 elements.
eb3irms7.inp: B32H elements.
ebo3ims7.inp: B32OS elements.
eboiims7.inp: B32OSH elements.
eb3arms7.inp: B33 elements.
eb3jrms7.inp: B33H elements.
ep22pms7.inp: PIPE21 elements.
inicurv_pipe2d_xpl.inp: PIPE21 elements in Abaqus/Explicit.
ep2hpms7.inp: PIPE21H elements.
ep23pms7.inp: PIPE22 elements.
ep2ipms7.inp: PIPE22H elements.
ep32pms7.inp: PIPE31 elements.
inicurv_pipe3d_xpl.inp: PIPE31 elements in Abaqus/Explicit.
ep3hpms7.inp: PIPE31H elements.
ep33pms7.inp: PIPE32 elements.
ep3ipms7.inp: PIPE32H elements.
ese4sms7.inp: S4 elements.
esf4sms7.inp: S4R elements.
es54sms7.inp: S4R5 elements.
es68sms7.inp: S8R elements.
es58sms7.inp: S8R5 elements.
es59sms7.inp: S9R5 elements.
es63sms7.inp: STRI3 elements.
es56sms7.inp: STRI65 elements.
esc8sms7.inp: SC8R elements.
esc8sms7_eh.inp: SC8R elements with enhanced hourglass control.
ecss8sms7.inp: CSS8 elements.

Fine mesh tests:

eb22rfs7.inp: B21 elements.
eb2hrfs7.inp: B21H elements.
eb32rfs7.inp: B31 elements.
eb3hrfs7.inp: B31H elements.
ebo2ifs7.inp: B31OS elements.
ebohifs7.inp: B31OSH elements.
ep22pfs7.inp: PIPE21 elements.
ep2hpfs7.inp: PIPE21H elements.
ep32pfs7.inp: PIPE31 elements.
ep3hpfs7.inp: PIPE31H elements.
ese4sfs7.inp: S4 elements.
esf4sfs7.inp: S4R elements.
es54sfs7.inp: S4R5 elements.
esc6sfs7.inp: SC6R elements.
esc8sfs7.inp: SC8R elements.
esc8sfs7_eh.inp: SC8R elements with enhanced hourglass control.
ecss8sfs7.inp: CSS8 elements.